Convergence Analysis of Online Regularized Statistical Learning in Reproducing Kernel Hilbert Space with Non-Stationary Data

מושגי ליבה

The core message of this paper is to establish the mean square consistency of an online regularized learning algorithm in reproducing kernel Hilbert space (RKHS) with dependent and non-stationary data streams. The authors introduce the concept of random Tikhonov regularization path, and show that if the regularization path is slowly time-varying, then the output of the algorithm is consistent with the regularization path in mean square. Furthermore, if the data streams satisfy the RKHS persistence of excitation condition, then the output of the algorithm is consistent with the unknown function in mean square.

תקציר

The paper studies the convergence of recursive regularized learning algorithms in the reproducing kernel Hilbert space (RKHS) with dependent and non-stationary online data streams.
Key highlights:

The authors introduce the concept of random Tikhonov regularization path, which involves randomly time-varying operators induced by the input data. This reframes the statistical learning problem as an ill-posed inverse problem with randomly time-varying forward operators.
The authors investigate the mean square asymptotic stability of two types of random difference equations in RKHS, where the non-homogeneous terms are respectively a martingale difference sequence and the drifts of the regularization paths.
The authors show that if the random Tikhonov regularization path is slowly time-varying, then the tracking error between the output of the algorithm and the regularization path tends to zero in mean square.
The authors introduce the RKHS persistence of excitation (PE) condition, which ensures that the random regularization path can approximate the unknown function. They prove that if the random regularization path is slowly time-varying and the data stream satisfies the RKHS PE condition, then the output of the algorithm is consistent with the unknown function in mean square.
For independent and non-identically distributed online data streams, the authors show that the algorithm achieves mean square consistency if the data-induced marginal probability measures are slowly time-varying and the average measure has a uniformly strictly positive lower bound, without the convergence assumption on the marginal probability measures and the priori information of the unknown function.

סטטיסטיקה

None.

ציטוטים

None.

תובנות מפתח מזוקקות מ:

Convergence Conditions of Online Regularized Statistical Learning in Reproducing Kernel Hilbert Space With Non-Stationary Data

by Xiwei Zhang,... ב- arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03211.pdf

שאלות מעמיקות

What are some potential applications of the proposed online regularized learning algorithm in RKHS with non-stationary data

The proposed online regularized learning algorithm in RKHS with non-stationary data has several potential applications across various fields. One key application could be in the field of financial forecasting, where the algorithm can be used to predict stock prices or market trends based on non-stationary data streams. Another application could be in the field of healthcare, where the algorithm can be utilized to analyze patient data and make predictions about disease progression or treatment outcomes. Additionally, the algorithm could be applied in the field of natural language processing for sentiment analysis or text classification tasks using non-stationary data. Overall, the algorithm's ability to handle non-stationary data streams makes it versatile and applicable in a wide range of real-world scenarios.

How can the RKHS persistence of excitation condition be verified or relaxed in practical scenarios

The RKHS persistence of excitation condition can be verified or relaxed in practical scenarios through various methods. One approach is to analyze the data streams and assess the level of variation and persistence over time. This can involve statistical tests or time series analysis techniques to determine if the condition is met. Additionally, incorporating domain knowledge and expertise in the specific application area can help in understanding the dynamics of the data and verifying the persistence of excitation condition. In practical scenarios where the condition may not be fully met, relaxation strategies such as adjusting the algorithm parameters or incorporating additional regularization techniques can be employed to ensure algorithm performance and stability.

Can the techniques developed in this paper be extended to other types of online learning algorithms, such as online gradient descent or online mirror descent, in the context of non-stationary data

The techniques developed in this paper for online regularized learning in RKHS with non-stationary data can be extended to other types of online learning algorithms, such as online gradient descent or online mirror descent. The key lies in adapting the regularization and update mechanisms to suit the specific characteristics of the algorithm while considering the non-stationary nature of the data. By incorporating similar concepts of regularization paths, tracking errors, and drift analysis, these algorithms can be enhanced to handle non-stationary data streams effectively. Additionally, leveraging the insights from this paper on mean square consistency and stability can guide the extension of these techniques to other online learning frameworks, ensuring robust performance in dynamic environments.